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I think "envy/jealousy" cannot explain the difference between 羡慕 and 妒忌. So far as I know, 羡慕 is a neutral verb, but 妒忌 has derogatory sense and its degree of envy is higher than 羡慕.
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StanOct 10 '13 at 1:33

This answer is perfectly correct, just too short for an answer. @Stan, what do you mean by 妒忌 having a derogatory sense?
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杨以轩Oct 10 '13 at 1:53

@QuestionOverflow: Hmm I find you're right. Sorry for my English-Chinese dictionary explains envy as "羡慕 and 妒忌", so I misunderstand English doesn't distinguish 羡慕 and 妒忌. The derogatory sense refers to the bad meaning of "妒忌/嫉妒: you have something good, so I hate you have that besides I hope I also can have that", it seems it's just like the definition of "jealousy".
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StanOct 10 '13 at 2:38

Synonymous (at least as far as I can tell) to 妒忌 are both 嫉妒 and 妒嫉. All three refer to jealousy.
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Janus Bahs JacquetDec 8 '13 at 16:18

here are some examples,
我羡慕你。 I envy you(maybe because you are good at singing.), which means that I hope to have the same thing that you have(I want to be as good at singing as you are.).

我嫉妒你。 I'm jealous of you(maybe because you are good at singing), the meaning overlaps with that of the previous sentence, but this sentence implies the probability that I might do something to stop you being that good, due to the fact that you are so good at singing, but I'm not as good as you.